Geometric proof that if n is a non-perfect square, then √n is irrational.

Question

I know there is a geometric proof of the irrationality of √2. I thought maybe this one could be generalized for √n when n is a non-perfect square, but I could not find something like that anywhere.

Does anyone know if such a geometric proof exist? I'm researching on different kinds of proof for this theorem, but could only find the algebraic ones.

Answer 1

I do not see a general solution.

However, this can be done for $\sqrt{n^2 + 1}$ and $\sqrt{n^2 - 1}$ given any integer $n > 1$.

You can find this result in TM Apostol's "Irrationality of The Square Root of Two -- A Geometric Proof" found here.